Off-center angle measurement system

ABSTRACT

An angle measurement system including a magnet coupled to a rotating member and adapted to provide a magnetic field which rotates with the rotating member about a rotational axis of the rotating member, and an integrated circuit angle sensor disposed within the magnetic field at a radially off-center position from the rotational axis. The integrated circuit angle sensor includes first and second bridges of magneto resistive elements configured to respectively provide first and second signals representative of substantially orthogonal first and second directional components of the magnetic field and together representative of an angular position of the rotating member, and a set of adjustment parameters for adjusting attributes of the first and second signals having values selected to minimize errors in the first and second signals.

BACKGROUND

Magnetic sensing devices, such as Giant Magnetoresistance (GMR) sensors,for example, are used in a wide variety of applications. For example,GMR sensors are often used for sensing the angular position of arotating member, such as a motor shaft. In such applications, apermanent magnet, sometimes referred to as a “pill”, is typicallymounted to or embedded in an end of the motor shaft and is centered onthe axis of rotation. The GMR elements or resistors, which are typicallyconnected to form one or more bridges, are also positioned so as to becentered on the axis of rotation, with the resulting output signalsgenerated by the GMR resistors being indicative of the angular positionof the rotating shaft.

However, for many reasons, such as space limitations, for example, itmay not always be possible or desirable to position an angle sensor,such as a GMR sensor at the end of the rotating shaft. In suchinstances, toothwheels or polewheels are sometimes mounted to therotating shaft and a magnetic sensor (e.g. a Hall effect sensor) arepositioned proximate to the wheel and detect the passing of each toothor pole to determined the angular position of the rotating shaft.However, while toothwheels and pole wheels are typically very accurate,they can be quite costly and provide only discrete (not continuous)location measurements as compared to magnetoresistive-based anglesensors, such as GMR type angle sensors, for example.

For these and other reasons, there is a need for the embodiments of thepresent disclosure.

SUMMARY

One embodiment provides an angle measurement system including a magnetcoupled to a rotating member and adapted to provide a magnetic fieldwhich rotates with the rotating member about a rotational axis of therotating member, and an integrated circuit angle sensor disposed withinthe magnetic field at a radially off-center position from the rotationalaxis. The integrated circuit angle sensor includes first and secondbridges of magneto resistive elements configured to respectively providefirst and second signals representative of substantially orthogonalfirst and second directional components of the magnetic field andtogether representative of an angular position of the rotating member,and a set of adjustment parameters for adjusting attributes of the firstand second signals having values selected to minimize errors in thefirst and second signals.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings are included to provide a furtherunderstanding of the present invention and are incorporated in andconstitute a part of this specification. The drawings illustrate theembodiments of the present invention and together with the descriptionserve to explain the principles of the invention. Other embodiments ofthe present invention and many of the intended advantages of the presentinvention will be readily appreciated as they become better understoodby reference to the following detailed description. The elements of thedrawings are not necessarily to scale relative to each other. Likereference numerals designate corresponding similar parts.

FIG. 1 is an end view of an off-center angle measurement systemaccording to one embodiment.

FIG. 2 is a side view of the off-center angle measurement system of FIG.1.

FIG. 3 a block diagram generally illustrating one embodiment of a GMRangle sensor suitable for use with the off-center angle measurementsystem of FIG. 1.

FIG. 4 is a schematic diagram generally illustrating a GMR resistorconfiguration according to one embodiment.

FIG. 5 is a graph illustrating examples of output curves provided by theGMR angle sensor of FIG. 3.

FIG. 6 is a diagram illustrating an example of a GMR sensor positionedat a center of rotation of a magnet according to one embodiment.

FIG. 7 is a graph of angle error from non-uniform magnetic field.

FIG. 8 is a graph of simulated error caused by adjusting amplitudeparameters.

FIG. 9 is a graph of simulated error caused by adjusting orthogonalityparameter.

FIG. 10 is graph of error caused by radial position relative to magnet.

FIG. 11 is schematic diagram of an array of GMR resistors according toone embodiment.

FIGS. 12A through 12C illustrate examples of GMR sensor bridgeconfigurations according to one embodiment.

FIG. 13 is a flow diagram illustrating an operating and calibrationprocedure according to one embodiment.

DETAILED DESCRIPTION

In the following Detailed Description, reference is made to theaccompanying drawings, which form a part hereof, and in which is shownby way of illustration specific embodiments in which the invention maybe practiced. In this regard, directional terminology, such as “top,”“bottom,” “front,” “back,” “leading,” “trailing,” etc., is used withreference to the orientation of the Figure(s) being described. Becausecomponents of embodiments of the present invention can be positioned ina number of different orientations, the directional terminology is usedfor purposes of illustration and is in no way limiting. It is to beunderstood that other embodiments may be utilized and structural orlogical changes may be made without departing from the scope of thepresent invention. The following detailed description, therefore, is notto be taken in a limiting sense, and the scope of the present inventionis defined by the appended claims.

FIGS. 1 and 2 respectively illustrate end and side views of anoff-center GMR angle-measurement system 30 according to one embodimentwhich, as will be described in greater detail below, employs a pluralityof adjustable GMR parameters to correct angle measurement errors causedby non-linear magnetic field lines. According to one embodiment,off-center angle measuring system 30 includes a permanent ring magnet32, divided equally into a north pole 34 (indicated by the “filled”area) and a south pole 36 (indicated by the “shaded” area), mounted on arotating member 38, such as a rotating motor shaft, and a GMR anglesensor 40. A magnetic field of ring magnet 32 extends from north pole 34to south pole 36, as indicated by magnetic field lines 42 a through 42d.

According to one embodiment, GMR angle sensor 40 comprises asemiconductor chip having a GMR resistor region 44 (i.e. a “sensitive”region) mounted on a printed circuit board (PCB) 46. According to oneembodiment, PCB 46 and thus, GMR angle sensor 40, are positioned in aplane 47 parallel to a surface 48 of ring magnet 32 such that resistorregion 44 is substantially centered along an x-axis 50 extending throughan axis of rotation 52 of rotating shaft 38 and between an inner radius(Ri) 54 and an outer radius (Ro) 56 of ring magnet 32. For purposes ofthe description herein, a 0-degree angular position of rotating shaft 38is defined as being along x-axis 50, as indicated at 58, a 90-degreeposition is defined as being along y-axis 60 (perpendicular to andextending through axis of rotation 52), as indicated at 62, and so on. Adirection of rotation of rotating shaft 38 is by rotational arrow 64.

FIG. 3 is a block diagram generally illustrating one embodiment of GMRangle sensor 40 when configured as a semiconductor chip. As illustrated,in addition to GMR resistor region 44, GMR angle sensor 40 furtherincludes a controller 70 and a memory 72, with memory 72 storing valuesfor a plurality of GMR parameters 74, which will be described in greaterdetail below. GMR angle sensor 40 further includes a plurality of PINs,such as a supply voltage (V_(DD)) pin 76, a ground pin 78, and a dataI/O pint 80.

FIG. 4 is a schematic diagram generally illustrating GMR resistor region44 according to one embodiment. As illustrated, GMR resistor region 44includes a pair of GMR sensor bridges 90 and 92, with sensor bridge 90formed by four GMR resistors 94 a-94 d and sensor bridge 92 formed byfour GMR resistors 96 a-96 d. According to the bridge implementation ofFIG. 4, GMR sensor bridges 90 and 92 are disposed orthogonal to oneanother and are respectively configured to sense an x-component and ay-component of a rotating electrical field, such as the magnetic fieldindicated by the dashed lines at 98.

A supply voltage V_(DD) 100 is applied to a terminal 102 via pin 76 andvoltage signals V_(X)+ and V_(X)− of GRM sensor bridge 90 are measuredat terminals 104 and 106, and voltage signals V_(Y)+ and V_(Y)− of GMRsensor bridge 92 are measured at terminals 108 and 110. In response toan external magnetic field, such as magnetic field 98, one or more ofthe GMR resistors 94 a-94 d and 96 a-96 d change in their electricalresistances, causing changes in voltage signals V_(X)+ and V_(X)− atterminals 104 and 106 and voltage signals V_(Y)+ and V_(Y)− at terminals108 and 110 which are representative of an angular position of magneticfield 98 relative to a reference vector (e.g. 0-degrees).

FIG. 5 is a graph 120 illustrating an “ideal” output signals V_(X)(COS)122 and V_(Y)(SIN) 124 provided by GMR angle sensor 40 basedrespectively on voltage signals V_(X)+ and V_(X)− and voltage signalsV_(Y)+ and V_(Y)− as magnetic field 98 rotates from 0-360 degrees, asindicated by the angle α at 128. As illustrated by the graph at 130,output signals V_(X)(COS) 122 and V_(Y)(SIN) 124 respectively representx- and y-components of a vector 132 indicative of the angular positionof magnetic field 98. According to one embodiment, output signalsV_(X)(COS) 122 and V_(Y)(SIN) 124 are provided to an external devicewhich determines the angular position of magnetic field 98 based on theabove relationship.

As mentioned above, graph 120 represents output signals V_(X)(COS) 122and V_(Y)(SIN) 124 under ideal conditions. Under such ideal conditions,each of the GMR resistors 94 a-94 d and 96 a-96 d are identical, and GMRsensor bridges 90 and 92 are perfectly orthogonal to one another, forexample, and output signals V_(X)(COS) 122 and V_(Y)(SIN) 124 provide anexact representation of the angular position of magnetic field 98.However, in reality, due to manufacturing tolerances, there aretypically inherent variations between each of the GMR resistors 94 a-94d and 96 a-96 d which can lead to amplitude and offset variation betweenoutput signals V_(X)(COS) 122 and V_(Y)(SIN) 124. Additionally, thedirections of fixed magnetic field portions of GMR resistors 94 a-94 dand 96 a-96 d are not likely to be perfectly orthogonal to one anotherwhich can phase errors between output signals V_(X)(COS) 122 andV_(Y)(SIN) 124. These amplitude, offset, and phase error, in-turn, leadto errors in the determined angular position of the magnetic field, suchas magnetic field 98.

According to one embodiment, GMR sensor 40 includes a plurality of GMRparameters which are employed, such as by controller 70 (see FIG. 3), toadjust and correct output signals V_(X)(COS) 122 and V_(Y)(SIN) 124 tocompensate for such variations. According to one embodiment, such GMRparameters are dynamically adjustable, such as via an input pin of GMRsensor 40, and values of GMR parameters 74 are stored in memory 72. SuchGMR parameters include adjustable amplitude and offset parameters foreach of the output signals V_(X)(COS) 122 and V_(Y)(SIN) 124, and anorthogonality parameter to adjust the phase of either V_(X)(COS) 122 orV_(Y)(SIN) 124. The amplitude and offset parameters are employed tosubstantially equalize the amplitudes of V_(X)(COS) 122 and V_(Y)(SIN)124, and the orthogonality parameter is adjusted to as to maintain asubstantially 90-degree phase shift between V_(X)(COS) 122 andV_(Y)(SIN) 124.

With reference to FIG. 1, one difficulty associated with positioning GMRangle sensor 40 off-center from permanent ring magnet 32 is thatmagnetic field lines at the perimeter of the magnet are bent or curved.In contrast, as illustrated by FIG. 6, when the GMR resistor region 44of GMR sensor 40 is centered on an axis of rotation of a permanentmagnet, such as that of an example pill-type permanent magnet 132embedded in an end of a rotating shaft, such as rotating shaft 38, themagnetic field lines passing through GMR resistor region 44 aresubstantially linear and uniform. However, as illustrated by FIG. 1,when GMR resistor region 44 of GMR sensor 40 is positioned off-centerfrom the axis of rotation of a permanent magnet, such as illustrated byFIG. 1, the magnetic field lines passing through GMR resistor region 44,such as magnetic field line 42 a, are curved and not uniformly spaced.This non-uniformity of the magnetic field can cause errors in outputsignals V_(X)(COS) 122 and V_(Y)(SIN) 124 generated by GMR angle sensor40 which, in-turn, cause errors in the measured angle of the rotatingshaft to which the permanent magnet is attached, such as rotating shaft38 to which permanent ring magnet 32 is coupled.

FIG. 7 is a graph 140 illustrating an example of an angle error in theangular position of rotating shaft 38 measured by GMR angle sensor 40 ofFIG. 1 and caused by the non-uniformity of the magnetic field at theperimeter of permanent ring magnet 32. As apparent from graph 140, theresulting angle error of curve is approximately sinusoidal in shape overa 360-degree rotation of the rotating shaft. It is noted, however, thaterrors in the measured angular position of rotating shaft 38 by GMRangle sensor 40 due to inherent amplitude and orthogonality variationsbetween GMR sensor bridges, such as GMR sensor bridges 90 and 92 (anddue to variations in the physical construction of GMR angle sensor 40)are also sinusoidal in nature.

For example, FIG. 8 is a graph 150 of a simulated error resulting froman amplitude difference between output signals V_(X)(COS) 122 andV_(Y)(SIN) 124. The simulation was performed by setting the GMRamplitude parameters to values which create a 1% difference in amplitudebetween V_(X)(COS) 122 and V_(Y)(SIN) 124 with no offset ororthogonality variations. As illustrated by graph 150, the resultingerror in the angle measured by GMR angle sensor 40 is sinusoidal innature over a 360-degree rotation of the magnetic field, with a 1%amplitude resulting in an error varying between 0 and 0.3%.

Similarly, FIG. 9 is a graph 160 of a simulated error resulting from aphase error between output signals V_(X)(COS) 122 and V_(Y)(SIN) 124.The simulation was performed by setting the orthogonality parameter to avalue which creates a 0.5-degree variation from a 90-degree phasedifference between V_(X)(COS) 122 and V_(Y)(SIN) 124 with no amplitudeor offset variations. Again, as illustrated by graph 160, the resultingerror in the angle measured by GMR angle sensor 140 is sinusoidal innature over a 360-degree rotation of the magnetic field, with a0.5-degree orthogonality error resulting in an error varying between 0and 0.5%.

According to one embodiment, based on the similar sinusoidal nature ofthe error in the angle measured by GMR angle sensor 40 resulting fromthe physical variations between the GMR resistors and GMR sensor bridgesand from the non-uniformity of the magnetic field, during a calibrationprocedure, the GMR parameters of a GMR angle sensor of an off-center GMRangle measuring system, such as GMR angle sensor 40 of off-center GMRangle measuring system 30 of FIG. 1, are adjusted to compensate forerrors resulting from the non-uniformity of the off-center magneticfield as well as for error due to amplitude and orthogonalityvariations.

According to one embodiment, during such a calibration procedure of anoff-center angle measurement system, such as off-center GMRangle-measurement system 30 of FIG. 1, rotating shaft 38 is driven sothat the angular position of rotating shaft 38 is at a known or actualposition during rotation, such as through use of a toothwheel orpolewheel based angle measurement system which is temporarily coupled torotating shaft 38. The angular position of rotating shaft 38 as measuredby GMR angle sensor 40 is then compared to the actual position and thevalues of GMR parameters 74 are adjusted until an error between theangular position of rotating shaft 38 as measured by GMR angle sensor 40and the actual position is substantially minimized. Again, by adjustingthe GMR parameters of GMR angle sensor 40 in this fashion, errors in theangular position as measured by GMR angle sensor 40 resulting from bothphysical variations of the GMR resistors and GMR sensor bridges and fromvariations due the non-uniformity of the off-center magnetic field aresubstantially corrected.

Due to the off-center variations in the magnetic field of the magnet,such as ring magnet 32, the accuracy of the angular position of rotatingshaft 38 as measured by GMR angle sensor 40 is also influenced by theposition of GMR angle sensor 40 between inner radius 56 and outer radius58 of permanent ring magnet 32. FIG. 10 is a graph illustrating examplesof the magnitude of output vector 142 resulting from GMR angle sensor 40output signals V_(X)(COS) 122 and V_(Y)(SIN) 124 when disposed atdifferent positions along x-axis 50 between inner radius 56 and outerradius 58 as shaft 38 rotates over a range of 90-degrees (i.e. 0-90degrees). In the illustrated example of FIG. 10, ring magnet 32 has aninner radius 56 of 3 mm, and outer radius of 8 mm, and a thickness of 3mm, with GMR angle sensor 40 being positioned 1.5 mm from surface 48.

Curves 172, 174, 176, 178, 180, 182, and 184 respectively illustrate themagnitude of output vector 142 when GMR angle sensor 40 is positioned atradii of 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, 9 mm, and 10 mm from a center ofring magnet 32. As illustrated by the curves of graph 170, the magnitudeof output vector 142 varies as ring magnet 32 rotates with shaft 38,which can cause errors in the angle measurement provided by GMR anglesensor 40. The errors in the angle measurement provided by GMR anglesensor 40 can be reduced by positioning GMR angle sensor at the radiallocation which results in the least variation in the magnitude of outputvector 142. In the example illustrated by FIG. 10, a radial position of7 mm from a center of ring magnet 32 provides the least amount ofvariation in magnitude of output vector 142 and represents the bestradial location at which to position GRM angle sensor 40.

In view of the above, according to one embodiment, during a calibrationprocedure of an off-center angle measurement system, such as off-centerGMR angle-measurement system 30 of FIG. 1, a radial position of GMRangle sensor 40 relative to ring magnet 32 is adjusted to determine thelocation which provides the least amount of variation in output vector142. Such a position corresponds to the position between inner radius 54and outer radius 56 at which the magnetic field of ring magnet 32 ismost uniform.

As illustrated by FIG. 11, according to one embodiment, GMR resistorregion 44 includes a first plurality of GMR resistors 190, including GMRresistors 194 a-194 h, having fixed magnetic field portions fixed alongan x-axis 192, for example (as indicated by the horizontal arrows inFIG. 11), and a second plurality of GMR resistors 192, including GMRresistors 196 a-196 h, having fixed magnetic field portions fixed alonga y-axis 193, for example (as indicated by the vertical arrows in FIG.11). The first plurality of GMR resistors 190 can be selectivelyinterconnected by an associated plurality of externally controllableswitches, as illustrated by switch 200, and the second plurality of GMRresistors 192 can be selectively interconnected by an associatedplurality of externally controllable switches, as illustrated by switch202, to form different GMR sensor bridge configurations (e.g. GMR sensorbridges 90 and 92 of FIG. 4) at different positions along x-axis 191.

For example, as illustrated by FIG. 12A, GMR resistors 194 a-194 d and196 a-196 d forming the left-most columns of GMR resistor region 44 arerespectively interconnected via the associated externally controllableswitches to form GMR sensor bridges 204 a and 206 a at a left-mostposition along x-axis 190 within GMR angle sensor 40. Similarly, asillustrated by FIG. 12B, GMR resistors 194 b, 194 d, 194 e, and 194 fand 196 b, 196 d, 196 e, and 196 f forming the center-most columns ofGMR resistor region 44 are respectively interconnected via theassociated externally controllable switches to form GMR sensor bridges204 b and 206 b at a central position along x-axis 190 within GMR anglesensor 40. Also, as illustrated by FIG. 12C, GMR resistors 194 e-194 hand 196 e-196 h forming the right-most columns of GMR resistor region 44are respectively interconnected via the associated externallycontrollable switches to form GMR sensor bridges 204 c and 206 c at aright-most position along x-axis 190 within GMR angle sensor 40.

By selecting which GMR resistors 94 a-94 h and 96 a-96 h are connectedto form the GRM sensor bridges, the radial position of the GMR sensorbridges of GMR angle sensor 40 relative to ring magnet 32 can befurther, and more finely, adjusted to determine a location whichprovides the least amount of variation in the magnitude of output vector142.

FIG. 13 is a flow diagram generally illustrating a process 220 forcalibrating and operating an off-center angle measurement systemaccording to one embodiment. Process 220 begins at 222 with the mountingof a magnet to a rotating shaft, such as rotating shaft 38, wherein themagnet provides a magnetic field which rotates with the shaft about arotational axis of the shaft. In one embodiment, the magnet comprises aring magnet, such as ring magnet 32, mounted about the shaft remote fromends of the shaft.

At 224, an integrated circuit magnetoresistive angle sensor, such as GMRangle sensor 40, is positioned in the magnetic field at a radiallyoff-center position from the axis of rotation. According to oneembodiment, the integrated circuit magnetoresistive angle sensor,includes first and second magnetoresistive sensing bridges respectivelyproviding first and second signals representative of orthogonal firstand second directional components of the magnetic field. According toone embodiment, the integrated circuit angle sensor includes a set ofadjustment parameters for adjusting attributes of the first and secondsignals. In one embodiment, the set of adjustment parameters includes afirst and a second amplitude parameter and a first and a second offsetparameter for respectively adjusting an amplitude and an offset of thefirst and second signals. In one embodiment, the set of adjustmentparameters further includes an offset parameter for one of the first andsecond signals to adjust a phase of the one of the first and secondsignals. In one embodiment, the first and second magnetoresistivesensing bridges comprises giant magnetoresistive elements. In oneembodiment, the first and second magnetoresistive sensing bridgescomprises anisotropic magnetoresistive elements.

At 226, a “known” angular position of the shaft is measured, such as byusing a discrete-type angle sensor, such as a conventional toothwheel-or polewheel-based angle sensor, for example, which is temporarilycoupled to the rotating shaft. At 228, the angular position of therotating as measured by the integrated circuit magnetoresistive anglesensor, the “measured” angular position, is determined from the firstand second signals.

At 230, the values of the set of adjustment parameters are adjusted tomodify attributes of the first and second signals to compensate forerrors in the measured angular position resulting from structuralinaccuracies between the magneto resistive elements of the first andsecond bridges and from non-uniformity of the magnetic field at theoff-center position until an error between the “known” angular positiondetermined at 226 and the “measured” angular position from determinedfrom the integrated circuit at 228 is substantially minimized.

According to one embodiment, process 220 is then complete, as indicatedat 232. However, according to one embodiment, as illustrated by thedashed lines at 234, process 220 additionally includes adjusting theradial position of the integrated circuit until variations in amagnitude of a vector based on the first and second signals aresubstantially minimized.

According to one embodiment, such as described above by FIGS. 11 and12A-12C, the integrated circuit magnetoresistive angle sensor includes afirst set of magnetoresistive elements and an associated first pluralityof switches which can be operated to selectively interconnect differentsubsets of the first set of magnetoresistive elements to form the firstmagnetoresistive sensing bridge at different positions within theintegrated circuit, and a second set of magnetoresistive elements and anassociated second plurality of switches which can be operated toselectively interconnect different subsets of the second set ofmagnetoresistive elements to form the second magnetoresistive sensingbridge at different positions within the integrated circuit.

According to one embodiment, as illustrated by the dashed lines at 236,process 220 includes adjustment of the radial position by operating thefirst and second pluralities of switches such that the resulting firstand second magnetoresistive sensing bridges are at positions within theintegrated circuit such that the variations in the magnitude of thevector are substantially minimized.

Although specific embodiments have been illustrated and describedherein, it will be appreciated by those of ordinary skill in the artthat a variety of alternate and/or equivalent implementations may besubstituted for the specific embodiments shown and described withoutdeparting from the scope of the present invention. This application isintended to cover any adaptations or variations of the specificembodiments discussed herein. Therefore, it is intended that thisinvention be limited only by the claims and the equivalents thereof.

1. An angle measurement system comprising: a magnet coupled to arotating member and providing a magnetic field which rotates with therotating member about a rotational axis of the rotating member; anintegrated circuit angle sensor disposed within the magnetic field at aradially off-center position from the rotational axis and including:first and second bridges of magneto resistive elements configured torespectively provide first and second signals representative ofsubstantially orthogonal first and second directional components of themagnetic field and together representative of an angular position of therotating member, wherein the first and second bridges are positionedadjacent to one another along a radius extending from the rotationalaxis and proximate to and in parallel with a surface of the magnet; anda set of adjustment parameters for adjusting attributes of the first andsecond signals having values selected to minimize errors in the firstand second signals.
 2. The angle measurement system of claim 1, whereinthe magnet comprises a ring magnet coupled about the rotating member ata position along shaft away from ends of the shaft.
 3. The anglemeasurement system of claim 1, wherein the set of adjustment parametershave values selected to minimize errors in the first and second signalscaused by non-uniformity of the magnetic field at the radiallyoff-center position.
 4. The angle measurement system of claim 1, whereinthe set of adjustment parameters includes a first and a second amplitudeparameter and a first and a second offset parameter for respectivelyadjusting an amplitude and an offset of the first and second signals. 5.The angle measurement system of claim 4, wherein the magnetoresistiveelements comprise anisotropic magnetoresistive elements.
 6. The anglemeasurement system of claim 4, wherein the set of adjustment parametersincludes an offset parameter for one of the first and second signals toadjust a phase of the one of the first and second signals.
 7. The anglemeasurement system of claim 6, wherein the magnetoresistive elementscomprise giant magnetoresistive elements.
 8. The angle measurementsystem of claim 1, wherein the integrated circuit is positioned at aradial distance from the axis of rotation which substantially minimizesvariations in a magnitude of a position vector obtained from the firstand second signals.